2015/12/21 by Mayeul Arminjon
Mathematics · #Advanced Topics in Algebra #Atlas (anatomy) #Constant curvature #Differentiable function #Equivalence (formal languages) #Geometric Analysis and Curvature Flows #Injective function #Manifold (fluid mechanics) #Mathematical Dynamics and Fractals #Pseudo-Riemannian manifold #Topological vector space #Vector field #Vector space #math.GM
paper · pdf · doi:10.1142/s0219887816500316
published as Int.J.Geom.Meth.Mod.Phys., 2016 · 38 pages. v3: version accepted for publication in Int. J. Geom. Meth. Mod. Phys.: stronger statements in Prop. 0 and Prop. 8, and precisions in the abstract, following from referee's suggestions; stronger form of Theorem 5; new examples
openalex publication_date 2015/12/21 · arxiv created 2015/12/22 · arxiv updated 2016/03/23 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
A global vector field [Formula: see text] on a “spacetime” differentiable manifold [Formula: see text], of dimension [Formula: see text], defines a congruence of world lines: the maximal integral curves of [Formula: see text], or orbits. The associated global space [Formula: see text] is the set of these orbits. A “[Formula: see text]-adapted” chart on [Formula: see text] is one for which the [Formula: see text] vector [Formula: see text] of the “spatial” coordinates remains constant on any orbit [Formula: see text]. We consider non-vanishing vector fields [Formula: see text] that have non-periodic orbits, each of which is a closed set. We prove transversality theorems relevant to such vector fields. Due to these results, it can be considered plausible that, for such a vector field, there exists in the neighborhood of any point [Formula: see text] a chart [Formula: see text] that is [Formula: see text]-adapted and “nice”, i.e. such that the mapping [Formula: see text] is injective — unless [Formula: see text] has some “pathological” character. This leads us to define a notion of “normal” vector field. For any such vector field, the mappings [Formula: see text] build an atlas of charts, thus providing [Formula: see text] with a canonical structure of differentiable manifold (when the topology defined on [Formula: see text] is Hausdorff, for which we give a sufficient condition met in important physical situations). Previously, a local space manifold [Formula: see text] had been associated with any “reference frame” [Formula: see text], defined as an equivalence class of charts. We show that, if [Formula: see text] is made of nice [Formula: see text]-adapted charts, [Formula: see text] is naturally identified with an open subset of the global space manifold [Formula: see text].